A certain antique watch increases in value by a constant percent each year. The watch was … — SAT

A certain antique watch increases in value by a constant percent each year. The watch was purchased for $150$ , and its value after $2$ years is $216$ . What is the annual percent increase in the value of the watch?

Cevap: 20 %

Cevap

The annual percent increase in the value of the watch is 20.

Because the watch increases in value by a constant percent each year, its value can be modeled by the exponential growth equation

V(t) = P(1+r)^t

, where

P

is the initial purchase price,

r

is the annual growth rate as a decimal, and

t

is the number of years. Substituting the given values

P = 150

t = 2

, and

V(2) = 216

yields

150(1+r)^2 = 216

. Dividing both sides by

150

gives

(1+r)^2 = 1.44

. Taking the positive square root of both sides results in

1+r = 1.2

, which simplifies to

r = 0.2

. Converting

0.2

to a percentage gives an annual percent increase of

20\%

. Therefore, the correct numeric answer is

20

Adım Adım Çözüm

Set up the exponential growth equation using the given values.

150(1+r)^2 = 216

Since the value increases by a constant percent each year, the growth is exponential and modeled by

V(t) = P(1+r)^t

Divide both sides of the equation by

150

to isolate the exponential base term.

(1+r)^2 = 1.44

Dividing

216

150

simplifies the equation to solve for the growth factor.

Take the positive square root of both sides of the equation.

1 + r = 1.2

Taking the square root of

1.44

isolates the expression

1+r

Solve for

r

and convert the decimal to a percentage.

r = 0.2

, which is

20\%

Subtracting

1

from both sides gives the annual rate

r = 0.2

, and multiplying by

100

gives the percent increase.

Anahtar Kavram

Solving for the growth rate in an exponential growth model

Alternatif Yöntem

Alternatively, since the question asks for a simple percent increase, you can test a few straightforward percentages. For example, if the annual increase were

10\%

, the value after 1 year would be

150 \times 1.10 = 165

, and after 2 years would be

165 \times 1.10 = 181.5

, which is too low. Testing

20\%

, the value after 1 year is

150 \times 1.20 = 180

, and after 2 years is

180 \times 1.20 = 216

. This matches the given value exactly.

Tahmini Süre:45s

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Adım Adım Çözüm

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